56 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA (the quotient of G by H) for which the canonical mapping n : G -> G/H is a homomorphism. (12.12.1) Let Gbea topo logicalgroup and Ha normal subgroup ofG. Then the quotient by H of the topology of G is compatible with the group structure of G/H. Let (x0,. y0) be a point of (G/H) x (G/H), and let x0 (resp. yQ) be an element of the coset x0 (resp. j)0). Every neighborhood of x0y0 is of the form 7c(U), where U is a neighborhood of x0 j>0 . There exists a neighborhood V of x0 and a neighborhood W of j0 in G such that the relations x e V and y e W imply xy e U. Hence the relations x e n(V) and y e n(W) imply xj; e 7r(U), and therefore the mapping (x, y)*-+xy is continuous at the point (x0, j>0). The proof of continuity of the map xN+x""1 is similar. Whenever we speak of the quotient G/H of a topological group G by a subgroup H, it is to be understood that G/H carries the quotient topology. (12.12.2) Let Gbea topological group and let H, K be two normal subgroups of G such that H ID K. Then the canonical bijection (p : G/K -»(G/H)/(K/H) is an isomorphism of topological groups. We know that (p is the unique mapping which makes the following diagram commutative: G-----------> G/K where n, n', and n" are the canonical homomorphisms. The definition of a quotient topology (12.10) shows that a subset U of (G/H)/(K/H) is open if and only if ri~\if-*(\3J) is open in G, that is, if and only if n'l((p"l(U)) is open in G; but this last condition signifies that ^-1(U) is open in G/K. Since <p is bijective, the result follows. (12.12.3) Let Gbea topological group, H a normal subgroup ofG, n: G->G/H the canonical mapping, and A a subgroup ofG. Then the canonical bijection of n(A) onto the quotient group AH/H is an isomorphism of topological groups. This is a particular case of (12.10.7), applied to the right action (h, X)H-»X/Z of H on G.